AMO - Advanced Modelling and Optimization, Volume 6, Number 1, 2004

Progress in Risk Measurement1

Siwei Cheng, Yanhui Liu and Shouyang Wang2

Institute of Systems ScienceAcademy of Mathematics and Systems Sciences

Chinese Academy of sciencesBeijing 100080, People’s Republic of China

Abstract: In this paper, we give the axiomatic characterization of riskmeasures and discuss the treads of developments in this area. The mainrecently proposed risk measures are presented, and their properties andrelations are discussed. The corresponding versions of dynamic risk mea-sure are also briefly introduced.

Keywords: Coherent risk measures; Value-at-Risk; Expected shortfall

1 Introduction

The term risk plays a pervasive role in the literature on economic, political, socialand technological issues. There are various attempts to define and to characterizethe risk for descriptive as well as prescriptive purpose (e.g., Brachinger and Weber[16] surveyed measures of perceived risk, regarding risk as a primitive). In thispaper, we mainly concentrate on financial risks and financial risk measures. Weassume that financial risks can be quantified on the basis of a random variable X.This random variable may present, for instance the future net worth of a position,the relative or absolute changes in values of an investment or the accumulated claimsover a period for a collective of insureds. In general, we regard risk as random profitor loss of a position. It can be positive (gains) as well as negative (losses). Pureloss situations can be analyzed by considering the random variable L=X− ≥ 0. Arisk measure is a mapping from the random variables representing risks to the realline. It gives a simple number that quantifies the risk exposure in a way that ismeaningful for the problem at hand.

Financial institutions, such as investment banks and insurance companies aroundthe globe are searching for techniques to enhance their risk management practice.

1Supported by NFSC, CAS and MADIS.2The corresponding author, Email: sywang@amss.ac.cn

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One of the critical steps is to construct a proper measure of risk. Possibly enhancedby the new trends in regulation of financial institutions and the reaction of theacademic community to the practical requirements, risk measurement is one of thefast evolving topics both in theoretical and practical fields in recent years.

The rest of paper is organized as follows. Section 2 is about new research devel-opments. Section 3 and section 4 are both about static risk measures. One containsthe axiomatic characterization of risk measures and the other discusses several typ-ical examples. Section 5 describes dynamic risk measures. And section 6 brieflyconcludes.

2 New research development

There has been a great momentum in research on risk measures in recent years,which has touched the following different, but interconnected aspects: 1) axiomaticcharacterization of risk measures; 2) construction of (coherent) risk measures; 3)premium principles in insurance context; 4) dynamic risk measures; 5) the relationbetween risk measures and other economic and financial theories. 6) application ofrisk measures to financial activities.

The first line of research was started by a group of scholar: Artzner, Delbaen,Eber, and Heath. The axiomatic definition of coherent risk measures was introducedin their path-breaking paper [8, 9]. They prescribed what mathematical properties ameaningful risk, more precisely capital requirement measures should have. Follmeret al. [35] and Frittelli et al. [38] proposed convex risk measures respectively. Basingon deviation measures, Rockafellar et al. [59] put forward expected-bounded riskmeasures. Their results will be presented in the next section of this paper.

In the pre-Markowitz era, financial risk was considered as a correcting factor ofexpected return. These primitive measures had the advantages of allowing an imme-diate preferential order of all investments. Variance was first proposed by Markowitzto measure the risk associated with the return of assets. Value-at-Risk (VaR) wasintroduced in 1994 by the leading bank—JP Morgan. It is very popular in practiceand has become part of financial regulations (Basel Committee on Banking Super-vision [13, 14]). Conditional Value-at-Risk (CVaR) has been proposed as a naturalremedy for the deficiencies of VaR, which is not a coherent risk measure in general.Many other kinds of risk measures are also constructed after the introduction ofcoherent risk measures. We will give some representative examples later.

Independently, practically at the same time, Wang, Young and Panjer presentedsimilar conclusions on a closely related subject related to insurance premium. Wanget al. [67] published the axioms for risk premium in a competitive insurance market.The most recent results of this line can be seen in Landsman and Sherris [49].

Dynamic risk measures have also been studied for evaluating multi-period risks.In real situations, risks are inherently multi-period due to intermediate cash flows

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caused by, for example, availability of extraneous cash, possible gains or losses be-cause of the changes of economic situations or adjustment of a portfolio. Having toaccount for intermediate cash flows is the essential difference between a single-periodrisk measure and a multi-period risk measure. Furthermore, dynamic risk measuresare also need to be considered for intermediate monitoring by supervisors and as-sessing the risk of a position over time. Follmer and Leukert [33], and Cvitanic etal. [23] have studied dynamic risk measures. Artzner et al. [11], Riedel [56], andBalbas [12] are all trying to define coherent risk measures for dynamic models. Weshall present some results in the section 5 of this paper.

Risk measures relate closely to utility function and asset pricing theories. Basedon the classical (µ, σ)−portfolio optimization theory of Markowitz (µ and σ refer toexpected return and standard deviation respectively), (µ, ρ)−portfolio optimizationwhere ρ is coherent risk measure can be considered. The (µ, ρ)−problem can betransformed into the problem of maximizing U=µ − λρ, where λ is a Lagrangianvariable. U can be interpreted as a utility function just when ρ is convex. Onthe other hand, when the preference function Φ(X) = E[U(X)], where U denotesthe utility function specific to each decision maker, do not separate risk or value,Jia and Dyer [43] proved that it is possible to derive an explicit risk measures:ρ(X) = −E[U(X − E(X))] for the specified utility function. Jaschke and Kuchler[42] proved that coherent risk measures are essentially equivalent to generalizedarbitrage bounds, named “good deal bounds” by Cerny and Hodges [21]. Thuscoherent risk measures link well with the established economic theories of arbitrageon one hand and utility maximization on the other hand. Delbaen [26] has relatedthe theory of risk measures to game theory and distorted probability measures. Wecan see Wang [67], and Carr [20] for other aspects in this line.

Risk measures have been widely applied to pricing, hedging, portfolio optimiza-tion, capital allocation and performance evaluation. To some extent, we can say thatthe application to performance measurement and capital allocation are the drivingforce to develop the risk theory. Since the end of 1996 in the European Unionand 1998 in the United States, the largest banks subjected to regulatory approvalhave been able to use the internal models to calculate VaR exposures for the trad-ing books, and thus the capital requirements for market risk to prevent insolvency.After coherent risk measures have been proposed, how to allocate risk capital byselecting proper risk measures become an important issue. When the risk measurefor a portfolio has been chosen, how to attribute risk contributions to subportfoliois another problem arising. This is of interest for risk diagnostics of a portfolio orfor performance analysis. See Tasche [63], Denault [28], Goovaerts et al. [39] andFischer [32] for further results. Other aspects of application can be found in Wang[67], Follmer and Leukert [33].

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3 Axiomatic characterization of risk measures

3.1 Coherent risk measures

Following the system of Artzner et al., we regard risks as future net worth. DenoteΩ the set of states of of nature, and assume it is finite; G the set of all risks, namelythe set of all real valued function on Ω. For simplicity, we will consider the marketmodels without interest rates in this section; it is immediate, however, to extendall the definitions and results to the situation with interest rate, by appropriatelydiscounting.

Definition 3.1 A map ρ : X → R will be called a coherent risk measure if itsatisfies the following axioms:

(a) Subadditivity: ρ(X + Y ) ≤ ρ(X) + ρ(Y ), ∀X, Y ∈ G;

(b) Positive homogeneity: if λ ≥ 0, then ρ(λX) = λρ(X);

(c) Monotonicity: if X ≤ Y , then ρ(X) ≥ ρ(Y );

(d) Translation invariance: if m ∈ R, then ρ(X + m) = ρ(X)−m.

Let us make some comments on the economic significance of these axioms.

Subadditivity (a): It has an easy interpretation. If the subadditivity did nothold, then ρ(X + Y ) ≥ ρ(X) + ρ(Y ). This would imply, for instance, that in orderto decrease risk, a firm might be motivated to break up into different incorporatedaffiliates. From the regulatory point of view, this would allow to reduce capitalrequirements. Note this axiom rules out the “semi-variance” type risk measuredefined by ρ(X) = −E(X) + σ2(X − E(X))−.

Positive homogeneity (b): We notice that subadditivity implies ρ(λX) ≤ λρ(X)for λ ≥ 0 and X ∈ G. Thus ρ(λX) ≥ λρ(X) is imposed by the positive homogeneityaxioms. This can be justified by liquidity considerations: a position (λX) could beless liquid, and therefore more risky, than that of λ smaller positions (X).

Monotonicity (c): It is obvious to expect that, if two final net worth are suchthat X ≤ Y , their risk measures have to satisfy ρ(X) ≥ ρ(Y ). This axiom rules outthe risk measure defined by ρ(X) = −E(X) + α · σ(X), where α > 0.

Translation invariance (d): Implies that the risk ρ(X) decrease by m, by addinga sure return m to a position X. Specially, we get

ρ(X + ρ(X)) = ρ(X)− ρ(X) = 0,

that is, when adding ρ(X) to the initial position X, we obtain a “neutral” position.

Typically, a coherent risk measure ρ can be represented by the supremum of theexpected negative of final net worth for some collection of “generalized scenarios”or probability measures P on states of the nature, i.e.,

ρ(X) = supp∈P

Ep[−X]

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The axioms of coherent risk measures have been very influential. These coher-ent risk measures can be used as (extra) capital requirements to regulate the riskassumed by market participants, traders, and insurance underwriters, as well as toallocate existing capitals. But we should realize that not all coherent risk measuresare reasonable to use under certain practical situations.

Coherent risk measures were extended in general spaces by Delbaen [26]. Laterwere extended to convex risk measures, also called weakly coherent risk measuresby relaxing the constraints of subadditivity and positive homogeneity, and insteadrequiring the following weaker condition:

Convexity (e): ρ(λX + (1− λ)Y ) ≤ λρ(X) + (1− λ)ρ(Y ), ∀λ ∈ [0, 1].

Definition 3.2 A map ρ : X → R will be called a convex risk measure if it satisfiesthe condition of convexity (e), monotonicity (c), and translation invariance (d).

Under the assumption ρ(0) = 0, Convexity of ρ implies that

ρ(λX) ≤ λ(X), ∀λ ∈ [0, 1], ∀X ∈ G,

ρ(λX) ≤ λ(X), ∀λ ≥ 1, ∀X ∈ G,

The second inequality suggests that when λ becomes large, the whole position (λX)is less liquid than λ singular position X. The first says when λ is small, the oppositeinequality must hold for certain reasons.

Convex risk measures take into account the situations where the risk of a positionincrease in a nonlinear way with the size of the position. They assure for secondorder stochastic dominance and has corresponding structure theorem:

ρ(X) = supP∈P

(EP [−X]− α(P )

),

where α : P → (−∞, ∞] satisfy α(P ) > −ρ(0) for any P ∈ P , and can be taken tobe convex and lower semicontinuous on P .

3.2 Expectation-bounded risk measures

We introduce deviation measures first.

Definition 3.3 A map ρ : X → R satisfying subadditivity (a), positive homogeneity(b), and the following two axioms are called deviation measures:

(f) Shift-invariance: ρ(X + m) = ρ(X), ∀X ∈ G, m ∈ R;

(g) Nonegative: ρ(X) ≥ 0, ∀X ∈ G.

We can see that ρ(X) > 0 for all nonconstant X, whereas ρ(X) = 0 for all constantX. Shift-invariance implies deviation measures is location-free. They fully measure

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the degree of uncertainty of X. Standard deviation and semi-standard deviation aretypical examples of this kind. Deviation measures and coherent risk measures arein fact mutually incompatible: there is no function can satisfy axioms (e) and (f)at the same time. The two kinds measure risk from different points of view: theformer regard risk as the magnitude of deviation, however, the latter quantify riskto determine the amount of capital has to be held as a cushion against potentialfuture losses.

Rockafellar et al. impose the conditions (a), (b), (d) and the following additionalcondition:

(h) ρ(X) > E(−X) for all nonconstant X, and ρ(X) = E(−X) for constant X.

Risk measures satisfying the above four conditions are called expectation-bounded.If monotonicity is further satisfied, we will have an expectation-bounded coherentrisk measures. The basic ideal of Rockafeller et al. is that applying a risk measurein Artzner et al. sense not to X itself, but to X−E(X) will induce a deviation mea-sure and vice versa. Coherent risk measure and deviation measure are therefore con-nected together. Formally there is a one-to-one correspondence between expectation-bounded risk measure and deviation risk measures. A simple examples for this corre-spondence is ρ1(X) = b·σ(X) for b > 0 and ρ2(X) = b·σ(X)−E(X). There exist riskmeasures both coherent and expectation-bounded. Take ρ(X) = −EP (X)+α·σ−P (X)

with 0 < α < 1, σ−P (X) =(E(max(E(X)−X, 0

)2)1/2

for example.

4 Some examples of risk measures

Variance and standard deviation have been traditional risk measures in economicsand finance since the pioneering work of Markowitz. The two risk measures exhibit anumber of nice technical properties. For example, the variance of a portfolio returnis the sum of the variance and covariance of the individual returns. Furthermore,variance can be used as a standard optimization function. Finally, there is a wellestablished statistical methods to estimate variance and covariance. However, vari-ance does not account for fat tails of the underlying distribution and therefore isinappropriate to describe the risk of low probability events, such as default risks.Secondly, variance penalizes ups and downs equally. Moreover, mean-variance deci-sions are usually not consistent with the expected utility approach, unless returnsare normally distributed or a quadratic utility function is chosen.

A general class of downside risk is the class of lower-partial-moment of degreek(k = 0, 1, 2, · · · ):

LPMk(c, X) = E[max(c−X, 0)k

],

or, in normalized form (k ≥ 2) : LPMk(c, X)1/k, where c denote a reference levelfrom which deviation are measured. It can be an arbitrary deterministic target oreven a stochastic benchmark.

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Let k = 1, we get expected regret:

ER(X) = E[max(c−X, 0)

],

It was utilized by Carino and Ziemba in the Russell Yasuda Kasai financial planningmodel. It is actually an average portfolio underperformance compared to a fixedtarget or some benchmark portfolio. ER can be computed by a linear programmingmodel based on scenario approach.

Let c = E(X), k = 1, we have lower-semi-absolute deviation :

E[max(E(X)−X, 0)

],

which is considered by Ogryczak and Ruszczynski [52] and Gotoh and Konno [40].

Let c = E(X), k = 2, we get semi-variance and semi-standard deviation.

In the following, I shall consider some other familiar risk measures: VaR, CVaR,expected shortfall (ES), and spectral risk measures in detail.

4.1 Value-at-Risk

VaR is a very easy and intuitive concept. It points out how much one may loseduring specified period (e.g. two weeks) with a given probability and how muchcapital should be set to control the risk exposure of a firm. VaR serves for thedetermination of the capital requirements that banks have to fulfill in order to backtheir trading activities.

We join here Delbaen [27] taking V aRα as the absolute value of the worst lossnot to be exceeded with a probability of at least 1− α.

Definex(α) = qα(X) = inf

x ∈ R : P [X ≤ x] ≥ α

as the lower α− quantile of X,

x(α) = qα(X) = infx ∈ R : P [X ≤ x] > α

as the upper α− quantile of X.

Its formal definition is:

V aRα(X) = −x(α) = q1−α(−X).

Let α ∈ (0, 1] be fixed. Consider the risk measure ρ given by

ρ(X) = V aRα(X),

Then ρ has the following properties:

(1) Monotonicity: if X ≤ Y , then ρ(X) ≥ ρ(Y );

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(2) Positive homogeneity: ρ(hX) = hρ(X), for h ≥ 0;

(3) Translation invariance: ρ(X + a) = ρ(X)− a, for a ∈ R;

(4) Law invariance: if P [X ≤ t] = P [Y ≤ t] for all t ∈ R, then ρ(X) = ρ(Y );

(5) Comonotonic additivity: ρ(f X + g X) = ρ(f X) + ρ(g X), for f, gnon-decreasing.

Law invariance is a crucial condition for a risk measure to be estimated fromempirical data and ensures that the measures are suitable for industrial applications.So it is a critical property for application. There have been many methods toestimate VaR (see Duffie and Pan [30] for an overview). Examples are historicalsimulation methods and J.P. Morgan’s RiskMetrics [46].

From shareholders’ or managements’ perspective, the quantile “VaR” at thecompany level is a meaningful risk measure since the default event itself is of primaryconcern, and the size of shortfall is only secondary. But VaR in general turns out tobe not even a convex measure and in particular not subadditive even when the tworandom variables are independent. This is its major drawback. The subadditivityplays a fundamental role in risk measurement, especially in the area of credit risk.Only in the very special case in which the joint distribution of return is elliptic, VaRis subadditive, i.e.,

V aRα(X + Y ) ≤ V aRα(X) + V aRα(Y ), X, Y ∈ GNon-subadditivity means that the risk of a portfolio may be larger than the sumof stand-along risks of its components. Thereupon it could happen that a welldiversified portfolio require more regulatory capital than a less diversified portfolio.Hence, managing risk by VaR may fail to stimulate diversification and it preventsto add up to the VaR of different risk sources.

VaR is a risk measure that only concerns about the frequency of defaults, notthe size of defaults. it is argued that VaR is an “all or nothing” risk measure. VaRmodels are usually based on the assumption of normal asset returns and will not workunder extreme price fluctuations. If an extreme event causes ruin occurs, there is nomore capital to cushion losses. Basak and Shapiro found that having embedded VaRinto an optimization framework, VaR risk managers incur large losses than non-riskmanagers in the most adverse states of the world.

Furthermore, it is inappropriate to use VaR in practice because of its non-convexity. It can have many local extremes, which will lead to unstable risk ranking.VaR is a model dependent risk measurement because, by definition, it depends onthe initial reference probability.

At the latest in 1999, when coherent risk measures appeared, it became clear thatVaR cannot be considered as an adequate risk measurement to allocate economiccapital for financial institutions. In spite of this, as a compact representation of risklevel, VaR has met the favor of regulatory agencies to measure downside risk andhas been embraced by corporate risk managers as an important tool in overall riskmanagement process.

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4.2 Expected shortfall

The term ES we used here stems from Acerbi et al. [2], despite the fact that in otherliteratures this term was already used sometimes in other meanings. Rockafellar andUryasev [57] have given the formal definitions of CV aR+, CV aR−, and CVaR. Theydistinguished the three clearly. In fact, CVaR in their paper is the same with ERhere. Expected shortfall is proposed as a remedy for the deficiencies of VaR andis characterized as the smallest coherent and law invariant risk measurement todominate VaR.

In simple words, ES at a specified level α is the average loss in the worst 100α%cases. It measures how much one can lose on averages in states beyond the V aRα

level. Assume E[X−] < ∞, The accurate definition of ESα is:

ESα(X) = −α−1(E[X1X≤x(α) + x(α)

(α− P [X ≤ x(α)]

)), (4.1)

where 1 is the indicator function

1A(a) =

1, a ∈ A,0, a 6∈ A.

When the underlying distribution is continuous, (4.1) will become simple:

ESα(X) = α−1E[X1X≤x(α)

].

We can prove that ES is a coherent risk measure. Further more,

ESα(X) = −α−1

∫ α

0

qu(X)du. (4.2)

Together with the properties of VaR, (4.2) implies that ES is law invariant andcomonotonic additive. It also shows ES is the coherent risk measure used in Kusuoka[48] as main building block for the representation of law invariant coherent riskmeasurements.

ESα is continuous with respect to α. Hence regardless of the underlying distri-butions, we can be sure that the risk measured by ESα will not change dramaticallywhen there is a switch in the confidence level, say some base points.

ES is monotonic function to α, that is

ESα+ε(X) ≤ ESα(X), ∀α ∈ (0, 1), ∀ε > 0, with α + ε < 1.

Another advantages of ES is that it can be estimated efficiently even in caseswhere the usual estimators for VaR fail. First define α−tail mean (TM) as:

TMα(X) = −ESα = α−1E[X1X≤x(α)

]= x(α).

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Then TMα is the average the worst 100α% cases. The tail mean is likely to benegative, and the ES represents potential loss as positive number in most cases.Indeed they are different names for the same object.

Let some independent samples (X1, X2, · · · , Xn) of X are given. Define theorder statistics X1:n ≤ · · · ≤ Xn:n as the sorted values of n-tuple (X1, X2, · · · , Xn).Approximate the number of 100α% elements in the sample by

[nα] = max

m∣∣ m ≤ nα, m ∈ N

.

The set of 100α% worst cases is therefore presented by the least [nα] outcomesX1:n, · · · , X[nα]:n

. The natural estimator for the expected loss in the 100α% worst

cases is therefore simply given by

TMnα (X) =

[nα]∑i=1

Xi:n

[nα].

We can get

limn→∞

[nα]∑i=1

Xi:n

[nα]= x(α) a.s. (4.3)

If X is integrable, then the convergence in (4.3) holds in L1 too.

More importantly, ES is a convex function with respect to positions, allowingthe construction of efficient optimizing algorithms. In particular, it has been shownthat ES can be minimizing using linear programming techniques, which makes manylarge-scale calculations practical. And moreover, such numerical calculation is effi-cient and stable.

ES is different from the tail conditional expectation (TCE) and worst conditionalexpectation (WCE) advanced by Artzner et al. [7]:

TCEα = −EX∣∣X ≤ x(α)

,

WCEα(X) = − infE[X|A] : A ∈ F , P (A) > α

.

TCEα is usually larger than the set of selected worst cases since X ≤ x(α)may happen to have a probability larger than 100α%. It is a coherent risk measureonly when restricted to continuous distributions while may violate subadditivity ongeneral distributions. The natural estimation for TCE is the negative of average ofall Xi ≤ x

(α)n , i.e.,

TCEαn (X) = −

n∑i=1

Xi1Xi≤X[nα]:n

n∑i=1

1Xi≤X[nα]:n

.

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WCE is a coherent risk measure but only useful in a theoretical setting since itdepends not only on the distribution of X but also on the structure of the under-lying probability space. Both TCE and WCE are sensible to small changes in theconfidence level α when applied to discontinuous distributions.

Comparing ES, TCE and WCE, we get

TCEα(X) ≤ WCEα(X) ≤ ESα(X).

ES is the maximum of WCEs when the underlying probability space varies. If thedistribution of X is continuous, we have

TCEα(X) = WCEα(X) = ESα(X).

ES closely relates to expected return (ER). Testuri and Uryasev [65] demon-strated the relationship between ER and ES. They formally prove that a portfolio,which minimizes ES, can be obtained by doing a sensitivity analysis with respectto the threshold in the expected regret. An optimal portfolio in ES sense is alsooptimal in the expected regret sense for some threshold in the regret function. Theinverse statement is also valid, i.e., if a portfolio minimizes the expected regret, thisportfolio can be found by doing a sensitivity analysis with respect to the ES con-fidence level. A portfolio, optimal in expected regret sense, is also optimal in ESsense for some confidence level.

When comparing expected shortfall with VaR, basing on Monte-Carlo simula-tion, Yamai and Yoshiba [73] get that expected shortfall is easily decomposed intorisk factors and optimized, while VaR is not, but expected shortfall requires a largersize of sample than VaR for the same level of accuracy. Risk decomposition enablesrisk managers to select assets that provide the best risk-return trade-off, and to al-locate economic capital to individual risk factors. Under market stress, Yamai andYoshiba [72] find that: first, VaR and expected shortfall may underestimate the riskof securities with fat-tailed properties and a high potential for large losses; second,VaR and expected shortfall may both disregard the tail dependence of asset returns;third, expected shortfall has less of a problem in disregarding the fat tails and thetail dependence than VaR does. The superintendent office of financial institutionsin Canada has put in regulation for the use of expected shortfall to determine thecapital requirement.

4.3 Spectral risk measure

Spectral risk measures Mφ(X) are defined by

Mφ(X) = −∫ 1

0

x(p)φ(p)dp, (4.4)

where φ ∈ L1([0, 1]). And φ is said to be an admissible risk spectrum if it is positive,decreasing and ‖φ‖ = 1.

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Mφ(X) is a coherent risk measure if and only if φ is an admissible risk spec-trum. φ can be regarded as a weight function reflecting an investor’s subjective riskaversion. This means there exists a one-to-one correspondence between risk aversionfunction φ and coherent spectral risk measures. The fact that φ(p) is decreasing pro-vides an intuitive insight of the coherence concept: a measure is coherent if it assignsbigger weight to worse cases. Introduce a measure dµ(α) on α ∈ [0, 1] satisfying thenormalization condition ∫ 1

0

αdµ(α) = 1.

We can get

Mφ(X) = −∫ 1

0

x(p)φ(p)dp =

∫ 1

0

ES(α)(X)dµ(α), (4.5)

Eq. (4.5) suggests spectral risk measure can be built by ESα.

Let φ(p) = 1α10≤p≤α, Then

ESα(X) = Mφ(X).

In this case, The weight function is simply uniform in p ∈ [0, α] and zero else-where. In general case, φ(p) assigns different weights to different “p-quantile slices”of the left tail.

It is not difficult to see that V aRα(X) can also be expressed as

V aRα(X) = Mφ(X) with φ(p) = δ(p− α),

where Dirac delta function is defined by∫ b

a

f(x)δ(x− c)dx = f(c), ∀c ∈ (a, b)

Spectral risk measure enjoys the properties of law-invariance and comonotonicadditivity. It is also a very simple object to be used in practice. We can estimateMφ by Mn

φ on a sample n i.i.d realizations X1, · · · , Xn of X. Xi;n has the samemeaning as before. Let

Mnφ (X) = −

n∑i=1

Xi:nφi,

where φi satisfies: (1)φi ≥ 0; (2)φi ≥ φj, if i < j; and (3)n∑

i=1

φi = 1.

Given φ(p) as a representative positive decreasing function with supp∈[0,1]

φ(p) < ∞,

the most natural choice when φi is

φi =φ(i/n)

n∑k=1

φ(k/n), i = 1, · · · , n

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If E[X+] < ∞ and E[X−] < ∞, we can prove that Mnφ is a consistent estimator:

converges to Mφ with probability 1 for n →∞.

From the perspective of portfolio optimization, Acerbi et al. [1] revealed thatrisk-reward problem is shown to coincide with the unconstrained optimization ofa single suitable spectral measure. In other words, minimizing a spectral measureturns out to be already an optimization process itself, where risk minimization andreturns maximization cannot be disentangled from each other.

Spectral risk measures have some similarities with distorted risk measures definedon distorted risk probabilities:

ρ(X) = E∗(X) = −∫ 0

−∞g(F (x)dx +

∫ ∞

0

[1− g(F (x)]dx,

where g : [0, 1] → [0, 1] is an increasing function with g(0) = 0 and g(1) = 1.Distortion function g can reflect the risk preference of an investor. Proper riskmeasures can be got by correctly choosing function g.

There are numerous risk measures. In the above, we just illustrate some typicalexamples.

5 Dynamic risk measure

Consider a finite time interval [0, T ] during which all economic activities take place,and suppose that the target is a liability C with maturity T . In a complete marketunder the no arbitrage assumption, consider an agent who cannot afford to commitat time t = 0 the entire amount

C(0) = E[ C

S0(T )

],

which would guarantee perfect hedging. Here S0(t) is the price of risk-free instrumentat time t in the market. The expectation E is calculated with respect to the unique,risk-neutral equivalent martingale measure. Then C is a risk that has to be properlymeasured.

Cvitanic and Karatzas [23] proposed measuring risk as the smallest expecteddiscounted net-loss:

ρ(x, C) = infπ(·)∈A(x)

E0

(C −Xx,π(T )

S0(T )

)+

, (5.1)

where x is the initial capital available at time t = 0; A(x) is the class of admissibleportfolio strategies; Xx,π(·)is the wealth process corresponding to portfolio π(·) andinitial capital x. Under certain conditions, explicit computation are got and it isthen straightforward to determine the smallest amount of capital that keeps the

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exposure to risk below a given, acceptable level. This is the dynamic version of thestatic lower-partial-moment risk proposed to amend the dynamic VaR version:

ρ(x, C) = supπ(·)∈A(x)

P0

[Xx,π(T ) ≥ C

].

which fails to take into account the magnitude of the net hedging loss(C−Xx,π(T )

)+.

(5.1) can be extended to the following form:

ρ(x, C) = infπ(·)∈A(x)

E0

[l(C −Xx,π(T )

S0(T )

)+],

where l is a concave function.

It is possible that the agent faces some uncertainty of the financial market itselfin addition to genuine risk that the liability C represents. Under this situation, wecapture the uncertainty by allowing for a suitable family P = Pνν∈D of “real worldprobability” or “scenarios” that are general enough to incorporate the uncertaintyabout the actual values of stock-appreciation rates, instead of just one (P0). Possiblemeasures are set to control the risk amount, which is bounded by a lower-measureof risk:

ρ(x, C) = supν∈D

infπ(·)∈A(x)

Eν

(C −Xx,π(T )

S0(T )

)+

, (5.2)

which corresponds to the maximal risk of the type (5.1) from the point of view ofan agent faced with the “worst possible scenario” ν ∈ D; an upper-measure of risk:

ρ(x, C) = infπ(·)∈A(x)

supν∈D

Eν

(C −Xx,π(T )

S0(T )

)+

, (5.3)

which is viewed by a regulator who regards the agent’s efforts merely as attempts to“contain the worst that can happen”. The quantity of (5.2) and (5.3) can be thoughtof as the lower (max-min) and upper (min-max) values of fictitious “stochastic gamebetween the agent and the market respectively. A saddle point of this game is shownto be the pair (π(·), ν), where π(·) corresponds to the investment strategy whichborrows the amount C(0) − x from the bank at time zero, and then invests in themarket according to the optimal hedging portfolio πC(·) for C, ν is the risk-neutralmeasures included in the possible “real world” measures. Coincidence of (5.2) and(5.3) means that the agent and regulator can reach agreement about how the riskassociated with the liability C is to be quantified.

In a stable hedging with finite probability space, x = 0, S(T ) = 1, and A(0) onlyconsisting of π(·) = 0, we have

ρ(C) = ρ(0, C) = supν∈D

Eν(C+).

14

The quantity of (5.2) satisfies the following properties:

(1) ρ(x, C) ≤ ‖(C − x)+‖∞ = ess sup(C(ω)− x)+;

(2) ρ(x1 + x2, C1 + C2) ≤ ρ(x1, C1) + ρ(x2, C2);

(3) ρ(λx, λC) = λρ(x, C), for λ ≥ 0;

(4) x 7−→ ρ(x, C) is convex increasing,

x 7−→ x + ρ(x, C) is convex increasing for fixed C.

These properties are related to single-period coherent risk measures.

Siu et al. [61] proposes Bayesian risk measures for derivatives which is easy to beimplemented and satisfies the four axioms of coherent risk measures. Their approachprovides investors with more flexibility in measuring risks of derivatives, by usingGerber Shiu’s option-pricing formula. The newly introduced concept—BayesianEsscher scenarios, takes both the subjective views and the market observation intoaccount.

It is more complex and difficult to consider dynamic risk measures. The corre-sponding systematic axioms that dynamic risk measures should satisfy have beentried to propose by lots of scholars. Each set of axiomatic characterizations hasits own reasonable part. But none becomes so influential as single-period coherentrisk measures. Multi-period models are still the subject of ongoing research. Theexisting ones characterize dynamic risk measures from different angle. Here I justillustrate two to bring some heuristics for further research.

Wang [70] proposed likelihood-based risk measures and gave the correspondingstructure form. This kind of risk measures satisfies six properties: continuity, riskseparability, consistency, stationarity, future independence and timing indifference.But Wang dees not assume translation invariance. Therefore the corresponding classof risk measures need not to be coherent nor convex.

Riedel [56] considered the changes of a position and availability of new infor-mation. Changes in the position are to be taken into account by recalculating the(stochastic) present value of future payments. Information is processed via updatingin a Bayesian way every single probability measure in the set of generalized scenar-ios. He assumed that a dynamic measures should satisfy the following conditions:independence of the past, adaptedness, monotone and predictable translation in-variance. In addition to these four conditions, a coherent risk measure should behomogenous and subadditivity. Riedel showed that every dynamic risk measure thatsatisfies the axioms of coherence, relevance and dynamic consistence can be repre-sented as the maximal expected present value of future losses where expectationsare taken with respect to a set of probability measures.

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6 Conclusion

The importance of a risk measure is in its ability to differentiate between differ-ent types of risk, its ability to accurately and consistently compare the severity ofdifferent risk portfolio, and its ability to be easily understood and applied. VaRdoes not take into account the severity of of an incurred damage event and failsto stimulate diversification. It will lead to disastrous results when used to measurerisk in most financial situations. As a response to these deficiencies, Artzner et al.first introduced the notion of coherent risk measures and put forward the axioms areasonable risk measure has to satisfy. This set of axioms has been widely acceptedand regarded as a landmark in the field of risk theory. Since then, researches onthe basis of coherent risk measures have been carried on. Convex and expectation-bounded risk measures were put forward. Expected shortfall is proposed as a naturalcoherent alternative to VaR. At the same time, it enjoys the properties of continuityand monotonicity in the confidence level. Moreover, it has efficient estimator andthe corresponding portfolio optimization problem can be solved by linear program-ming methods. Other different classes of (coherent) risk measures with their ownproperties are also constructed.

There also arise lots of related problems, for example, the relationship betweencoherent risk measures and other financial theories and how to choose proper riskmeasures in practice when we have so many choices. While optimizing a portfolioor allocating risk capital by companies as well as by regulators, we will want toknown which is the best: variance, VaR or some other coherent risk measures, therelationship of efficient frontiers obtained by solving the portfolio selection prob-lem under these measures or the principle of allocating capital among portfolios.Goovaerts et al. [39] examine properties of risk measures that can be consideredto be in line with some “best practice” rules in insurance, and argue that so-calledcoherent risk measures lead to problems. Financial risk include market risk, creditrisk, operation risk and so on. Each kind of risk has its own distinctive characters.Furthermore, newer financial instruments surface in the capital market every day.It is still great challenges to properly and effectively measure these risks. Comparedto single-period risk measures, dynamic risk measures are more complicate and dif-ficult. The corresponding axiomatic characterizations are still an ongoing research.On the fast developing topic of risk measurement, much work has already been done,however, more further researches still need to be carried on.

References

[1] Acerbi, C., Spectral measures of risk: a coherent representation of subjectiverisk aversion, Journal of Banking & Finance, 26(2002), 1505-1518.

[2] Acerbi, C., Nordio, C., and Sirtori, C., Expected shortfall as a tool for financialrisk management, Working paper, http://www.gloriamundi.org, 2001.

16

[3] Acerbi, C., and Simonetti P., Portfolio optimization with spectral measures ofrisk, Working paper, http://www.icer.it/workshop, 2002.

[4] Acerbi, C., and Tasche, D., On the coherence of expected shortfall, Journal ofBanking & Finance, 26(2002), 1487-1503.

[5] Alexander, C. (Ed.), Mastering risk, Financial Times Prentice Hall, London,2001.

[6] Andersson, F., Mausser, H., Rosen, D., and Uryasev, S., Credit risk optimiza-tion with conditional value-at-risk criterion, Mathematical Programming, Ser.B 89(2001), 273-291.

[7] Artzner, P., Application of coherent risk measures to capital requirements ininsurance, North American Actuarial Journal, 3(1999), 11-25.

[8] Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D., Thinking coherently,Risk, 10(1997), 68-71.

[9] Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D., Coherent measures ofrisk, Mathematical Finance, 9(1999), 203-228.

[10] Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D., Risk manage-ment and capital allocation with coherent measures of risk, Working paper,http://www.math.ethz.ch/finance, 2000.

[11] Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D., Ku, H., Coherent multi-period risk adjusted values, http:// www.math.ethz.ch/finance, 2001.

[12] Balbas, A., Garrido J., and Mayoral S., Coherent risk measures in a dynamicframework, Working paper, http://pascal.iseg.utl.pt/ cemapre, 2002.

[13] Basel Committee on Banking Supervision, Capital requirements and bank be-havior: the impact of the Basel accord, Working Paper, BIS, Basel, 1, 1999.

[14] Basel Committee on Banking Supervision, The new Basel capital accord, con-sultative document, BIS, Basel, January, 2001.

[15] Bell, D., Risk, return and utility, Management science, 41(1995), 23-30.

[16] Brachinger, H. W., and Weber, M., Risk as a primitive: a survey of measuresof perceived risk, Operations Research - Spektrum, 19(1997), 235-250.

[17] Brachinger, H. W. Measurement of risk. In: Derigs, U. (Ed.), Optimization andOperations Research, Eolss publishers Co Ltd, 2002, 1119-1137.

[18] Bredow, H. R., Value at risk, expected shortfall, and marginal risk contribution,www.wifak.uni-wuerzburg.de/wilan/wifak/bwl, 2002.

[19] Butsic, R. P., Solvency measurement for risk-based capital apphcat ons, Journalof Risk and Insurance, 61(1994), 656-690.

[20] Carr, P., Geman, H., and Madan, D., Pricing and hedging in incomplete mar-kets, Journal of Financial Economics, 62(2001), 131-168.

[21] Cerny, A., Hodges, S., The theory of good-deal pricing in financial markets,Imperial College Management School, London, 2000.

[22] Crouhy, M., Galai, D., and Mark, R., Risk management, McGraw-Hill, NewYork, 2001.

17

[23] Cvitanic, J., and Karatzas L., On dynamic measures of risk, Finance andStochastics, 3(1999), 451-482.

[24] Danielsson, J., The emperor has no clothes: limits to risk modelling, Journalof Banking & Finance, 26(2002), 1273-1296.

[25] Danielsson, J., Jorgensen, B.N., and de Vries, C.G., Incentives for effective riskmanagement, Journal of Banking & Finance, 26(2002), 1407-1425.

[26] Delbaen, F., Coherent measures of risk on general probability spaces, Workingpapers, http://math.ethz.ch/ delbaen, 2000

[27] Delbaen, F., PISA lecture notes, preprint, ETH Zurich,http://www.math.ethz.ch/ delbaen, 2000.

[28] Denault, M., Coherent allocation of risk capital, Journal of Risk, 4(2001), 1-34.

[29] Dhaene, J., Goovaerts, M.J., and Kaas, R., Economic capital allocation derivedfrom risk meausres, to appear in: North America Actuarial Journal, 7(2003).

[30] Duffie, D., and J. Pan, An overview of the value at risk, Journal of Derivatives,4(1997), 7-49.

[31] Fischer, T., Coherent risk measures depending on higher moments, Workingpaper, Universitat Heidelberg, http://www.rzuser.uni-heidelberg.de, 2001.

[32] Fischer, T., Risk capital allocation by coherent risk measures based on one-sidedmoments, To appear in insurance: Mathematics and Economics, 2002.

[33] Follmer, H., and Leukert P., Quantile hedging, Finance and Stochatics, 3(1999),251-273.

[34] Follmer, H., and Leukert P., Efficient hedging: cost versus shortfall risk, Fi-nance and Stochastics, 4(2000), 117-146.

[35] Follmer, H., and Schied, A., Convex measures of risk and trading constraints,Finance and Stochastics, 6(2002), 429-447.

[36] Frey, R., and McNeil, A., VaR and expected shortfall in portfolios of dependentcredit risks: conceptual and practical insights, Journal of Banking & Finance,26(2002), 1317-1334.

[37] Frittelli, M., Representing sublinear risk measures and pricing rules, Mimeo,July, http://www.dimequant.unimib.it, 2000.

[38] Frittelli, M., and Rosazza G, E., Putting order in risk measures, Journal ofBanking & Finance, 26(2002), 1473-1486.

[39] Goovaerts, M.J., Dhaene, J., and Kaas, R., Risk measures, measures for in-solvency risk and economic capital allocation, Tijdschrift voor Economie enManagement, XLVI, 4(2001), 545-559.

[40] Gotoh, J., Konno, H., Third degree stochatics domonance and mean risk anal-ysis, Management Science, 46(2000), 289-301.

[41] Gourieroux, C., Laurent, J.P., and Scaillet, O., Sensitivity analysis of values atrisk, Journal of Empirical Finance, 7(2000), 225-245.

[42] Jaschke S., and Kuchler U., Coherent risk measures and good-deal bounds,Fiance and Stochastics, 5(2001), 181-200.

18

[43] Jia, J., and Dyer, S., A standard measure of risk and risk-value models, Man-agement Science, 42(1996), 1691-1705.

[44] Jones, D., Emerging problems with the Basel capital accord: regulatory capitalarbitrage and related issues, Journal of Banking & Finance, 24(2000), 35-58.

[45] Jorion, P., VaR: the new benchmark for managing financial risk, McGraw Hill,New York, 2000.

[46] J. P. Morgan Co., Riskmetrics-Technical Document, 4th ed, New York, 1996.

[47] Konno, H., and Yamazaki, H., Mean-absolute deviation portfolio optimisa-tion model and its application to Tokyo stock market, Management Science,37(1991), 519-531.

[48] Kusuoka, S., On law invariant coherent risk measures, In: Advances in Mathe-matical Economics, Springer, Tokyo, 3(2001), 83-95.

[49] Landsman, Z., and Sherris, M., Risk measures and insurance premium princi-ples, Insurance: Mathematics and Economics, 29(2001), 103-115.

[50] Luciano, E., and Marena, M., Value at risk bounds for portfolios of non-normalreturns, Working paper, http://math.econ.unito.it, 2001.

[51] Manganelli, S., and Engle, R.F., Value at risk models in finance, Working paper,http://www.gloriamundi.org/var, 2001.

[52] Ogryczak, W., Ruszczynski, From stochastic dominance to mean-risk mod-els: semideviation as risk measures, European Journal of Operational Research,116(1999), 33-50.

[53] Panjer, H., Measurement of risk, solvency requirements and allocation of capitalwithin financial conglomerates, Research report 01-15, Institute of Insuranceand Pension Research, University of Waterloo, http://www.soa.org/research,2002.

[54] Pflug, G., Some remarks on the value-at-risk and the conditional value-at-risk, In: Uryasev, S. (Ed.), Probabilistic Constrained Optimization:Methodology and Applications, Kluwer Academic Publishers, Dordrecht,http://www.gloriamundi.org, 2000.

[55] Phelan, M.J., Probability and statistics applied to the practice of financialrisk management: the case of JP Morgan’s RiskMetrics, Journal of FinancialServices Research, 12(1997), 175-200.

[56] Riedel, F., Dynamic coherent risk measires, Working paper,http://www.wiwi.hu-berlin.de/ riedel, 2003.

[57] Rockafellar, R. T., and Uryasev, S., Conditional value-at-risk for general lossdistributions, Journal of Banking & Finance, 26(2002), 1443-1471.

[58] Rockafellar, R. T., and Uryasev, S., Optimization of conditional Valule-at-Risk,Journal of Risk, 2(2000), 21-41.

[59] Rockafellar, R., Uryasev, S., and Zabarankin, M., Deviation measures in riskanalysis and optimization, Research report, http://www.ise.ufl.edu/uryasev,2002.

19

[60] Silvapulle, P., and Granger, C.W., Large returns, conditional portfolio diversi-fication: Value-at-risk approach, Quantitative Finance, 1(2001), 542-551.

[61] Siu T. K., Tong H., and Yang H., Bayesian risk measures for derivatives viarandom esscher trandform, North American Actuarial Journal, 5(2001), 78-91.

[62] Szego, G., Measures of risk, Journal of Banking & Finance, 26(2002), 1253-1272.

[63] Tasche, D., Risk contributions and performance measurement, working paper,TU Munchen, http://www-m4.mathematik.tu-muenchen.de, 2000.

[64] Tasche D., Expected shortfall and beyond, Journal of Banking & Finance, 26(2002), 1519-1533.

[65] Testuri, C.E., and Uryasev, S., On relation between expected regretand conditional value-at-risk, Mimeo, ISE University of Florida, August,http://www.ise.u..edu/uryasev/testuri.pdf, 2000.

[66] Topaloglou, N., Vladimirou H., and Zenios S. A., CVaR models with selectivehedging for international asset allocation, Journal of Banking & Finance, 26(2002), 1535-1561.

[67] Wang, S. S., A class of distortion oprators for financial and insurance risks,Journal of risk and insurance, 67(2000), 15-36.

[68] Wang, S. S., Young, V.R., and Panjer, H.H., Axiomatic characterization ofinsurance prices, Insurance: Mathematics and Economics, 21(1997), 173-183.

[69] Wang, S. S., A risk measure that goes beyond coherence, Research re-port, Institute of Insurance and Pension Research, University of Walterloo,http://www.stats.uwaterloo.ca, 2002.

[70] Wang, T. A class of dynamic risk measures, Working paper, Faculty of Com-merce and Business Administration, University of British Columbia, Canada,1999.

[71] Wirch, J. L., Raising value at risk, North American Actuarial Journal, 3(1999),106-115.

[72] Yamai, Y., and Yoshiba, T., On the validity of value-at-risk: comparative analy-sis with expected shortfall, Institute for Monetary and Economics Studies, Bankof Japan, 20(2002), 57-85.

[73] Yamai, Y., and Yoshiba, T., Comparative analyses of expected shortfall andvalue-at-risk: their estimation error, decomposition, and optimization, Institutefor Monetary and Economics Studies, Bank of Japan, 20(2002), 87-121.

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